Transcription
I have one last--I was going to say, trick property. One last logarithm property to show you. So let me pick a suitably festive color for this last property. So let's say that just, I don't know, x to the n is equal to a. Nothing fancy there.
Well, that's just another way of saying that log base x of a is equal to n, right? That's the exact same--this is just the exact same way of writing the exact same thing. One's a logarithm, one's an exponent, right? These imply the same thing. But what we can do is, if n is actually equal to this expression, we can, like I did a couple of videos ago, you could just substitute this for n. So we could write x to this thing, log base x a. And we could set that as equal to what? a. Fascinating.
So now what I'm going to do and, actually, this is going to get pretty messy, is I'm going to raise--actually, let me write this a little more space. Undo. Oh, I can't keep undoing. Anyway, so let me write down here with more space. Because I'm going to do something fancy. So, ignore this. So, if I set x to the log base x of a, that equals--and you'll see why I'm giving you so much space right now--equals a. Now, what I want to do is, I want to raise both sides of this equation to 1 over this exponent. So I'm going to raise that to 1 over log base x of a. If I do something to one side of the equation, I have to do it to the other. So that's also, that's equal to a, to 1 over log base x to a.
I know, this is quite daunting already. But you'll see where I'm going. And hopefully nothing I've done is completely not-intuitive, right? This expression is just another way of writing this expression. And I substituted it for n. And now I'm raising both to this exponent. And you'll see why I'm doing that. Well, if you're raising something to an exponent and then you're raising that to an exponent, you just multiply the two, right? So they cancel out. Because this will be the numerator. And this'll be the denominator. So that gets us to this: x to the 1 power, right? Because log base x of a over log base x of a is equal to 1. So that's the same thing as x is equal to a to the 1 over log base x of a.
You're probably saying, Sal, where are you going with this. And I will sort of show you shortly. So, we could also just replace a with another variable, right? I could also write x is also equal to b to the 1 over log base x of b, right? Nothing strange there. The same exact thing I did with a, I could do with a. The same thing I did with a, I could do with b. So I've written these two expressions. I said x is equal to both of these things. So let's set them equal to each other. So, we know that a to 1 over log base x of a, is equal to b to the 1 over log base x of b. So, what can we do now? Well, let's raise both of these--actually, I'm running out of so much space. Let me clear this and go to the next page, or go to another page. Clear image. Invert.
So what did I just write? I said that, because I need a lot of space for what I plan to do. So, I said, a to the 1 over log base x of a--well, that equals b to the 1 over log base x of b. And hopefully you're satisfied with that. Now, let's raise both of these sides to the log base x of b power. This log base x's of b power. Now, hopefully you'll see why I'm doing this. On this side they'll cancel out, right? Because this becomes a numerator, that's the denominator. And on this side, you get a to the--this becomes the numerator, right, because we just multiply the exponents. Log base x, that little dot is an x. Of b over log base x of a. And what does that equal? Well, that equals just b, right? Because this over this is 1. This b to the 1. That equals b.
Now let's write this entire thing as a logarithm. a to this thing is equal to b. That's the exact same thing as saying that the logarithm base a of b is equal to this thing. Is equal to the log base x of b divided by the log base x of a. This might seem confusing, it might seem daunting, but we're actually going to do a lot of examples with this. And this is probably the single most useful identity, I guess you could call it, if you're using a calculator. Why? Because your calculator only has two bases. It either has log base, you know, base 10, or base e, right? And most of them, when you press the log button on your calculator, it assumes log base 10. So if I gave you a problem where I wanted to know what is the log base 7 of 3, right? Who knows? 7 to what power is 3? And there's no easy way, on most calculators, to do this. Well, you can use this identity. That this is the same thing as the log base 10 of 3, divided by the log base 10 of 7. And these are very easy to calculate on your calculator. You just type 3 and press log. It'll give you this number. And you press 7 and click on log, it'll give you this number. And then you're done. So hopefully you're satisfied that this is true and you have a little bit of an intuition of how to use it. And I'll make a bunch of videos now, on actually how you can use these logarithm properties. I just wanted to get it out of the way so that you're satisfied that they are true. I'll see you soon.